Riemann Hypothesis and strongly Ramanujan complexes from GL(n)

Abstract

We investigate the Riemann Hypothesis on combinatorial zeta functions associated to finite quotients of the affine building of GL(n). We prove that if the quotient complex is strongly Ramanujan then these zeta functions satisfy the Riemann Hypothesis. On the other hand, we show that the converse statement is also true provided the extra generic condition. In the end, we give an example to show that this generic condition is indeed necessary. (C) 2015 Elsevier Inc. All rights reserved.